Controlling Time: Weapons of Math Destruction

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The potential interest in a broadcast network grows proportionally to the number of users. For example, if a radio station doubles its audience, its audience value doubles.†Interest = K + S(n), where K represents fixed assets (e.g., for land and equipment), n is the number of users, and S is a coefficient named here for broadcast pioneer David Sarnoff.

What if there was a way to build interest more quickly?

A broadcast network reaches each user from a single source. In the 1980s, Bob Metcalfe noticed that a person-to-person transaction network has an additional component of interest proportional to the number of individual connections that can be made between users. If all such connections are equally interesting, we have Metcalfe’s Law(*): Interest = K + S(n) + M(n^2)

Can interest grow even faster?

Around the turn of the millennium, half a decade before first paper on what would become Teleplace, David Reed pointed out that group-forming networks have yet another component of interest that is proportional to the number of different sub-groups that can be formed. Reed’s Law is: Interest = K + S(n) + M(n^2) + R(2^n)

Each of the successive formulation is describing a theoretical potential. Monetizing a network into a successful business is not given by mere formula. Nonetheless, there is a lot that can be understood by looking at the mathematical properties of Reed’s Law.

We see that that as a network grows in size, we pass through three distinct regions where different things are important:

  1. If n, M, and R are small enough, all potential interest is linear with the number of users. More or better content is generally considered the key to interest in this region, where Content is King.
  2. Even with small M, the M term is eventually going to overtake the S term. When n is large enough, there is a transition point after which interest is dominated by the number of connections that can be made, which is quadratic with the number of users. In this region, Communication is King.
  3. If growth continues, the R term overtakes all, and interest is dominated by the number of groups that can be formed among the users. Growth is exponential in this region, in which Community is King.


We’ve seen the content vs communication regions play out many times. For example, person to person communication networks such as the Telephone system are going to grow faster than broadcast networks once they get large enough. If we have two separate, equal sized phone networks, their total value is just twice the value of one alone. But when they merge, the number of connections within the system – and hence the potential value – is four times the value of one alone. This suggests that there would be a tendency for cell phone networks to merge, and this is exactly what we have seen. Indeed, The Telephone Company grew to become the world’s most valuable company until it was broken up, after which the sum of its parts became less valuable than the whole. We have also seen this transition on the Internet. In the early days, before the WWW, the small number of Internet sites were dominated by file transfer FTP sites. Content was King. There were several mostly incompatible email systems, with addresses using characters we don’t see any more, such as % and !. Once we reached critical mass with the name@site addressing, email’s SMTP traffic eclipsed FTP and continued growing much faster from that point on. On the Web, once commerce transaction sites such as Amazon got large enough, they continued to grow much faster than news and informational sites.

eBay did not grow so much merely because it is an auction site. There were other transaction sites empowering a single company’s auctions. eBay was special because it allowed any number of people to instantly form their own communities around new auction items. Once there is enough users, the value of community grows much faster than even communication value. †While other search engines offered connections to content based on the content of those sites (titles, bolds, and other html tags on the page), Google has always really been about the communities of people interested in the many subjects embodied by queries. In addition to its community-based PageRank algorithm, Google revenue products are built around the context of what people are searching for rather than the content on a page. Compared with maintaining a mailing list, today’s social media sites allow users to fluidly join and quit an impossibly large set of ever-changing communities. And indeed, social media has been adopted much faster than email.

This is all very exciting, but the formula also tells us that there are several things to be wary of.

First of all, just as there is interest that comes from content, communications, and community, there are also costs that can grow linearly, quadratically, or even exponentially. Today’s social media sites control these costs with lots of limitations. You can have a group, but you can’t actually talk to everyone by voice or share live content. The potential interest of today’s community-oriented services does grow exponentially with the number of users, but the coefficients R, M, and S are incredibly small. There is fast growth, but the actual value is lower than it could be if one could deliver a non-trivial shared experience.

Second, the formulae have some qualifiers such as “all connections being equally interesting.” Consider a screen sharing application in which one application is shown at a time, and only the group leader can “drive” it, rather than letting anyone interact with everything. Andrew Odylyzko and other researchers point out that such asymmetric communications might have potential value growth that is n(log(n)) at best. That still leaves three growth regions, but with slower growth if we don’t pay attention to such qualifiers as symmetric use.

Finally, we note that not every system has a half billion users. Even a company with 100,000 employees cannot form nearly as many subgroups as Twitter. Suppose that a company uses a social media system with similar trivial capability as Twitter, but which is cut off from reaching folks outside of the company’s employees. With Twitter’s low R coefficient, the system isn’t going to be nearly as valuable to the company as Twitter is to the world a large. Indeed, it may be considerably less important than email or file sharing. Similarly, the company’s asymmetric app-sharing tools may also disappoint. Conversely, imagine if that company can have a system with a large S value for content, a large M value for communication, and a large R value for community.

*: The original forms of Metcalfe’s Law and Reed’s Law, and the original form of this article, used the term “value” instead of “interest.” I have edited this to use the term “interest”, as described here

Next: The Dial-tone for Cyberspace


About Stearns

Howard Stearns works at High Fidelity, Inc., creating the metaverse. Mr. Stearns has a quarter century experience in systems engineering, applications consulting, and management of advanced software technologies. He was the technical lead of University of Wisconsin's Croquet project, an ambitious project convened by computing pioneer Alan Kay to transform collaboration through 3D graphics and real-time, persistent shared spaces. The CAD integration products Mr. Stearns created for expert system pioneer ICAD set the market standard through IPO and acquisition by Oracle. The embedded systems he wrote helped transform the industrial diamond market. In the early 2000s, Mr. Stearns was named Technology Strategist for Curl, the only startup founded by WWW pioneer Tim Berners-Lee. An expert on programming languages and operating systems, Mr. Stearns created the Eclipse commercial Common Lisp programming implementation. Mr. Stearns has two degrees from M.I.T., and has directed family businesses in early childhood education and publishing.

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